Finite Group Databasebeta

← #27277638

Subgroups — #27277638

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274703C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272594C212yesmaximal, complemented
4#20272582C212yesmaximal, complemented
5C53×#2349C211yesmaximal, characteristic
6C2×#13133132—21no—
7C2×C22⋊#88322C2212yes—
8#16810483C2212yes—
9#16810390—21no—
10#16810349—21no—
11#16810347—21no—
12C2×C4×D4×C53—21no—
13C2×C4×D4×C53C2211yescharacteristic
14C53×#389—21no—
15C53×#385—21no—
16C53×#385—21no—
17C53×#382C411yescharacteristic, complemented
18C53×#380C411yescharacteristic, complemented
19C53×#380C2211yescharacteristic
20C4×C4⋊C4×C53C411yescharacteristic, complemented
21C2×#95×C53C411yescharacteristic, complemented
22C2×#95×C53C2211yescharacteristic
23C22×#88322—22no—
24C2×C4⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26#13133132—42no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133124—22no—
30C22×D4×C53D411yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53—21no—
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53C2311yescharacteristic
35C4×D4×C53—22no—
36C4×D4×C53—22no—
37C4×D4×C53—42no—
38C4×D4×C53—42no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C42×C53—21no—
49C2×C42×C53C2×C411yescharacteristic
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53C2×C411yescharacteristic
54#95×C53C2×C411yescharacteristic
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53D812yes—
62C2×D4×C53SD1612yes—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—42no—
70C22×C4×C53—42no—
71C22×C4×C53—42no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—81no—
75C22×C4×C53—81no—
76C22×C4×C53—81no—
77C22×C4×C53—81no—
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53#4112yes—
84C22×C4×C53#4112yes—
85C22×Dic53—42no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—42no—
89C4⋊C4×C53—81no—
90C4⋊C4×C53—81no—
91C22⋊C4×C53—22no—
92C22⋊C4×C53—22no—
93C22⋊C4×C53—42no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C42×C53—21no—
97C42×C53—21no—
98C42×C53—42no—
99C22⋊Dic53—42no—
100C22⋊Dic53—42no—
101C4⋊Dic53—82no—
102C2×#88322—42no—
103C2×#88322—82no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53#10212yes—
111C23×C53C2×C22⋊C411yescharacteristic
112C23×C53#12312yes—
113C23×C53C2×D811yescharacteristic
114C23×C53C2×SD1611yescharacteristic
115C23×C53#13712yes—
116D4×C53—24no—
117D4×C53—24no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—81no—
141C2×C4×C53—81no—
142C2×C4×C53—82no—
143C2×C4×C53—82no—
144C2×C4×C53—84no—
145C2×C4×C53#10212yes—
146C2×C4×C53#11711yescharacteristic
147C2×C4×C53#11711yescharacteristic
148C2×C4×C53#12312yes—
149C2×Dic53—82no—
150C2×Dic53—82no—
151C2×Dic53—82no—
152#88322—82no—
153#8760—531nomaximal, Sylow
154C22×C53—22no—
155C22×C53—21no—
156C22×C53—22no—
157C22×C53—21no—
158C22×C53—22no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—42no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53#38111yescharacteristic
167C22×C53C2×#10211yescharacteristic
168C22×C53#41911yescharacteristic
169C22×C53#48012yes—
170C22×C53#48312yes—
171C4×C53—22no—
172C4×C53—22no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—42no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53—81no—
179C4×C53—81no—
180C4×C53—81no—
181Dic53—162no—
182#2349D5311yescharacteristic
183#1799—531no—
184#1459—531no—
185#1294—532no—
186#1289—532no—
187C2×C53—22no—
188C2×C53—21no—
189C2×C53—21no—
190C2×C53—22no—
191C2×C53—42no—
192C2×C53#182311yescharacteristic
193C2×C53#182711yescharacteristic
194C2×C53#182811yescharacteristic
195C2×C4×D4—21no—
196C2×C4×D4C2×D5311yescharacteristic
197C2×#99—1061no—
198C2×C22⋊C8—532no—
199#389—21no—
200#385—21no—
201#385—21no—
202#382Dic5311yescharacteristic, complemented
203#380—532no—
204#380Dic5311yescharacteristic, complemented
205#380C2×D5311yescharacteristic
206C4×C4⋊C4Dic5311yescharacteristic, complemented
207C2×#95Dic5311yescharacteristic, complemented
208C2×#95C2×D5311yescharacteristic
209#342—1061no—
210#324—1061no—
211#323—1061no—
212C53#876011yescharacteristic, complemented, Sylow
213C22×D4C22⋊D5311yescharacteristic
214C23×C4—21no—
215C23×C4—21no—
216C23×C4C2×Dic5311yescharacteristic
217C23×C4C22×D5311yescharacteristic
218C22×C8—1062no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—42no—
222C4×D4—42no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—1062no—
228C2×C4⋊C4C22⋊D5311yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—1062no—
232C2×C22⋊C4C22⋊D5311yescharacteristic
233C2×C22⋊C4C22⋊D5311yescharacteristic
234C2×C42—21no—
235C2×C42C2×Dic5311yescharacteristic
236#99—2122no—
237C22⋊C8—1062no—
238C22⋊C8—1062no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—1062no—
243#95C2×Dic5311yescharacteristic
244#95C2×Dic5311yescharacteristic
245#95C2×Dic5311yescharacteristic
246#95C2×Dic5311yescharacteristic
247C24C22⋊Dic5311yescharacteristic
248C24C2×C22⋊D5311yescharacteristic
249C2×D4—22no—
250C2×D4—22no—
251C2×D4#1171992212yes—
252C2×D4#1171992312yes—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—2122no—
269C22×C4C22⋊Dic5311yescharacteristic
270C22×C4C22⋊Dic5311yescharacteristic
271C22×C4C22⋊Dic5311yescharacteristic
272C22×C4#1171994512yes—
273C22×C4#1171994512yes—
274C22×C4C22×Dic5311yescharacteristic
275C22×C4C2×C22⋊D5311yescharacteristic
276C2×C8—4242no—
277C2×C8—2122no—
278C4⋊C4—22no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—81no—
282C4⋊C4—81no—
283C4⋊C4—4242no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—42no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—2122no—
290C22⋊C4—2122no—
291C42—21no—
292C42—21no—
293C42—42no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23#1313313812yes—
301C23C2×C22⋊Dic5311yescharacteristic
302C23#1313320012yes—
303C23C2×#1171992211yescharacteristic
304C23C2×#1171992311yescharacteristic
305C23#1313324812yes—
306D4—24no—
307D4—24no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—84no—
335C2×C4—4242no—
336C2×C4—4242no—
337C2×C4—4242no—
338C2×C4#1313313812yes—
339C2×C4#1313317611yescharacteristic
340C2×C4#1313317611yescharacteristic
341C2×C4#1313320012yes—
342C8—4242no—
343C22—21no—
344C22—22no—
345C22—21no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22#1681048611yescharacteristic
356C22C2×#1313313811yescharacteristic
357C22#1681063211yescharacteristic
358C22#1681093612yes—
359C22#1681095312yes—
360C4—22no—
361C4—22no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—8482no—
371C2—21no—
372C2—22no—
373C2—21no—
374C2—22no—
375C2—42no—
376C2#2027480111yescharacteristic
377C2#2027481911yescharacteristic
378C2#2027482311yescharacteristic
subgroupG/H
centerC22#16810486
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2349C2
socleC22×C53#381
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2349D53
hypocenterC53#8760
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2349

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20272594C2semidirect
4. #20272582C2semidirect
17. C53×#382C4semidirect
18. C53×#380C4semidirect
20. C4×C4⋊C4×C53C4semidirect
21. C2×#95×C53C4semidirect
202. #382Dic53semidirect
204. #380Dic53semidirect
206. C4×C4⋊C4Dic53semidirect
207. C2×#95Dic53semidirect
212. C53#8760semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202747032
#202730332
#202725942
#202725822
C53×#23492
#876053